L(s) = 1 | + (0.745 − 0.745i)2-s + 0.887i·4-s + (−3.80 + 1.01i)5-s + (0.148 − 2.64i)7-s + (2.15 + 2.15i)8-s + (−2.07 + 3.59i)10-s + (0.913 − 0.244i)11-s + (−0.783 − 3.51i)13-s + (−1.85 − 2.08i)14-s + 1.43·16-s − 7.96·17-s + (0.451 − 1.68i)19-s + (−0.904 − 3.37i)20-s + (0.499 − 0.864i)22-s − 6.93i·23-s + ⋯ |
L(s) = 1 | + (0.527 − 0.527i)2-s + 0.443i·4-s + (−1.70 + 0.455i)5-s + (0.0562 − 0.998i)7-s + (0.761 + 0.761i)8-s + (−0.656 + 1.13i)10-s + (0.275 − 0.0738i)11-s + (−0.217 − 0.976i)13-s + (−0.496 − 0.556i)14-s + 0.359·16-s − 1.93·17-s + (0.103 − 0.386i)19-s + (−0.202 − 0.754i)20-s + (0.106 − 0.184i)22-s − 1.44i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.166292 - 0.608070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.166292 - 0.608070i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.148 + 2.64i)T \) |
| 13 | \( 1 + (0.783 + 3.51i)T \) |
good | 2 | \( 1 + (-0.745 + 0.745i)T - 2iT^{2} \) |
| 5 | \( 1 + (3.80 - 1.01i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.913 + 0.244i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 + (-0.451 + 1.68i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 6.93iT - 23T^{2} \) |
| 29 | \( 1 + (1.71 + 2.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.17 - 4.37i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (6.88 + 6.88i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.117 + 0.437i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.0936 - 0.0540i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.19 + 4.45i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.747 + 1.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.42 + 3.42i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.73 - 3.30i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.17 - 4.36i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.68 - 10.0i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.95 - 1.05i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.473 + 0.820i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.26 - 8.26i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.79 - 3.79i)T - 89iT^{2} \) |
| 97 | \( 1 + (11.5 - 3.09i)T + (84.0 - 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39702877606266362166550399603, −8.724300531437974794297220171196, −8.158149118346315845233236240300, −7.24176391794540666961138649905, −6.79207189043476719793517932060, −4.89158669224558384040790351494, −4.15879738695381818821780322647, −3.57188044264323567267584418968, −2.50838965473405391006269052568, −0.25667236959355315848973044428,
1.76507942801691314658182993128, 3.57534741387174055306378727613, 4.47116294906616294973157881745, 5.07475277504116248347325256124, 6.27760040759310410244901709707, 7.05679709704209107937659312264, 7.902068689611796333369602035625, 8.918845198679949800820354450200, 9.415919730458555935128577586353, 10.87704834759178297242958427409